Quantum solution for the onedimensional Coulomb problem
Abstract
The onedimensional hydrogen atom has been a much studied system with a wide range of applications. Since the pioneering work of Loudon [R. Loudon, Am. J. Phys. 27, 649 (1959).], a number of different features related to the nature of the eigenfunctions have been found. However, many of the claims made throughout the years in this regard are not correctsuch as the existence of only odd eigenstates or of an infinite bindingenergy ground state. We explicitly show that the onedimensional hydrogen atom does not admit a ground state of infinite binding energy and that the onedimensional Coulomb potential is not its own supersymmetric partner. Furthermore, we argue that at the root of many such false claims lies the omission of a superselection rule that effectively separates the right side from the left side of the singularity of the Coulomb potential.
 Authors:
 Departamento de Fisica, Universidad Autonoma Metropolitana, Unidad Iztapalapa, Apartado Postal 55534, Iztapalapa CP 09340 D. F. (Mexico)
 (Mexico)
 Publication Date:
 OSTI Identifier:
 21550229
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Physical Review. A; Journal Volume: 83; Journal Issue: 6; Other Information: DOI: 10.1103/PhysRevA.83.064101; (c) 2011 American Institute of Physics
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 74 ATOMIC AND MOLECULAR PHYSICS; ATOMS; BINDING ENERGY; COULOMB FIELD; EIGENFUNCTIONS; EIGENSTATES; HYDROGEN; MATHEMATICAL SOLUTIONS; ONEDIMENSIONAL CALCULATIONS; QUANTUM MECHANICS; SINGULARITY; SUPERSELECTION RULES; SUPERSYMMETRY; ELECTRIC FIELDS; ELEMENTS; ENERGY; FUNCTIONS; MECHANICS; NONMETALS; SELECTION RULES; SYMMETRY
Citation Formats
NunezYepez, H. N., SalasBrito, A. L., Solis, Didier A., Departamento de Ciencias Basicas, Universidad Autonoma Metropolitana, Unidad Azcapotzalco, Apartado Postal 21267, Coyoacan CP 04000 D. F., and Facultad de Matematicas, Universidad Autonoma de Yucatan, Periferico Norte Tablaje C. 13615, Merida, Yucatan. Quantum solution for the onedimensional Coulomb problem. United States: N. p., 2011.
Web. doi:10.1103/PHYSREVA.83.064101.
NunezYepez, H. N., SalasBrito, A. L., Solis, Didier A., Departamento de Ciencias Basicas, Universidad Autonoma Metropolitana, Unidad Azcapotzalco, Apartado Postal 21267, Coyoacan CP 04000 D. F., & Facultad de Matematicas, Universidad Autonoma de Yucatan, Periferico Norte Tablaje C. 13615, Merida, Yucatan. Quantum solution for the onedimensional Coulomb problem. United States. doi:10.1103/PHYSREVA.83.064101.
NunezYepez, H. N., SalasBrito, A. L., Solis, Didier A., Departamento de Ciencias Basicas, Universidad Autonoma Metropolitana, Unidad Azcapotzalco, Apartado Postal 21267, Coyoacan CP 04000 D. F., and Facultad de Matematicas, Universidad Autonoma de Yucatan, Periferico Norte Tablaje C. 13615, Merida, Yucatan. 2011.
"Quantum solution for the onedimensional Coulomb problem". United States.
doi:10.1103/PHYSREVA.83.064101.
@article{osti_21550229,
title = {Quantum solution for the onedimensional Coulomb problem},
author = {NunezYepez, H. N. and SalasBrito, A. L. and Solis, Didier A. and Departamento de Ciencias Basicas, Universidad Autonoma Metropolitana, Unidad Azcapotzalco, Apartado Postal 21267, Coyoacan CP 04000 D. F. and Facultad de Matematicas, Universidad Autonoma de Yucatan, Periferico Norte Tablaje C. 13615, Merida, Yucatan},
abstractNote = {The onedimensional hydrogen atom has been a much studied system with a wide range of applications. Since the pioneering work of Loudon [R. Loudon, Am. J. Phys. 27, 649 (1959).], a number of different features related to the nature of the eigenfunctions have been found. However, many of the claims made throughout the years in this regard are not correctsuch as the existence of only odd eigenstates or of an infinite bindingenergy ground state. We explicitly show that the onedimensional hydrogen atom does not admit a ground state of infinite binding energy and that the onedimensional Coulomb potential is not its own supersymmetric partner. Furthermore, we argue that at the root of many such false claims lies the omission of a superselection rule that effectively separates the right side from the left side of the singularity of the Coulomb potential.},
doi = {10.1103/PHYSREVA.83.064101},
journal = {Physical Review. A},
number = 6,
volume = 83,
place = {United States},
year = 2011,
month = 6
}

We have developed and tested in terms of atomic calculations an exact, analytic and computationally simple procedure for determining the functional derivative of the exchange energy with respect to the density in the implementation of the Kohn Sham formulation of density functional theory (KSDFT), providing an analytic, closedform solution of the selfinteraction problem in KSDFT. We demonstrate the efficacy of our method through groundstate calculations of the exchange potential and energy for atomic He and Be atoms, and comparisons with experiment and the results obtained within the optimized effective potential (OEP) method.

Analytic solution of the relativistic Coulomb problem for a spinless Salpeter equation
We construct an analytic solution to the spinless Swave Salpeter equation for two quarks interacting via a Coulomb potential, (2(del/sup 2/+m/sup 2/)/sup 1/2/M..cap alpha../r) psi(r) = 0, by transforming the momentumspace form of the equation into a mapping or boundaryvalue problem for analytic functions. The principal part of the threedimensional wave function is identical to the solution of a onedimensional Salpeter equation found by one of us and discussed here. The remainder of the wave function can be constructed by the iterative solution of an inhomogeneous singular integral equation. We show that the exact boundstate eigenvalues for the Coulomb problemmore » 
Analog of the FeynmanGellMann equation and an exact solution of the relativistic Coulomb problem for a charged spin3/2 particle
An exotic atom in which an ..cap omega../sup / hyperon is bound to a nucleus by a Coulomb field is considered. It is proposed to describe the hyperon by a 3rdrank symmetric spinor which obeys an equation of the FeynmanGellMann type. A formula is obtained for the energy levels of such a system. 
Computationally simple, analytic, closed form solution of the Coulomb selfinteraction problem in Kohn–Sham density functional theory
We have developed and tested in terms of atomic calculations an exact, analytic and computationally simple procedure for determining the functional derivative of the exchange energy with respect to the density in the implementation of the Kohn–Sham formulation of density functional theory (KSDFT), providing an analytic, closedform solution of the selfinteraction problem in KSDFT. We demonstrate the efficacy of our method through groundstate calculations of the exchange potential and energy for atomic He and Be atoms, and comparisons with experiment and the results obtained within the optimized effective potential (OEP) method.